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{{Expert needed |mathematics |talk=Expert needed |reason=There may be logic or math errors, complicated by the fact that this article was translated from a [[:de:Axonometrie|German original]] |date=May 2017}}
{{short description|The process of projecting a three-dimensional object onto a two-dimensional plane}}
[[File:Axonometrie-torbogen-c.svg|thumb|[[Cavalier projection]] of a semicircular archway.]]
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'''Axonometry''' is a graphical procedure belonging to [[descriptive geometry]] that generates a [[Planar graph|planar image]] of a three-dimensional object. The term "axonometry" means "to measure along [[Cartesian coordinate system|axes]]", and indicates that the [[Dimension|dimensions]] and [[Scaling (geometry)|scaling]] of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled [[parallel projection]] of the object. In general, the resulting parallel projection is [[oblique projection|oblique]] (the rays are not [[perpendicular]] to the image plane); but in special cases the result is [[orthographic projection|orthographic]] (the rays are perpendicular to the image plane), which in this context is called an '''[[Orthogonality|orthogonal]] axonometry'''.

In [[technical drawing]] and in [[architecture]], axonometric perspective is a form of [[Two-dimensional space|two-dimensional]] representation of [[Three-dimensional space|three-dimensional]] objects whose goal is to preserve the impression of [[volume]] or [[relief]]. Sometimes also called rapid perspective or artificial perspective, it differs from [[Cone|conical]] perspective and does not represent what the eye actually sees: in particular parallel lines remain parallel and distant objects are not reduced in size. It can be considered a conical perspective conique whose center has been pushed out to infinity, i.e. very far from the object observed.

The term ''axonometry'' is used both for the graphical procedure described below, as well as the image ''produced'' by this procedure.
 
''Axonometry'' should not be confused with ''[[axonometric projection]]'', which in English literature usually refers to ''orthogonal axonometry''.

== Principle of axonometry ==
[[File:Axonom-def.svg|thumb|The point <math>\overline P</math> is the projection of a point <math>P=(x,y,z)</math> onto the projection plane ''Π''. The foreshortenings are <math>v_x</math>, <math>v_y</math> and <math>v_z</math>.]]
[[Pohlke's theorem]] is the basis for the following procedure to construct a scaled parallel projection of a three-dimensional object:{{sfn|Graf|1961|p=144}}{{sfn|Stärk|1978|p=156}}

# Select projections of the coordinate axes, such that all three coordinate axes are not collapsed to a single point or line. Usually the z-axis is vertical.
# Select for these projections the [[foreshortening]]s, <math>v_x</math>, <math>v_y</math> and <math>v_z</math>, where <math>v_x,v_y,v_z > 0.</math>
# The projection <math>\overline P</math> of a point <math>P=(x,y,z)</math> is determined in three sub-steps (the result is independent of the order of these sub-steps):
#* starting at the point <math>\overline O</math>, move by the amount <math>v_x\cdot x</math> in the direction of <math>\overline x</math>, then
#* move by the amount <math>v_y\cdot y</math> in the direction of <math>\overline y</math>, then
#* move by the amount <math>v_z\cdot z</math> in the direction of <math>\overline z</math> and finally
# Mark the final position as point <math>\overline P</math>.

In order to obtain undistorted results, select the projections of the axes and foreshortenings carefully (see below). In order to produce an [[orthographic projection]], only the projections of the coordinate axes are freely selected; the foreshortenings are fixed (see [[:de:orthogonale Axonometrie]]).{{sfn|Graf|1961|p=145}}

== The choice of the images of the axes and the foreshortenings ==

[[File:Axo-parameter-gen.svg|thumb|Parameters and notations]]

'''Notation:'''
* <math>\alpha:</math> angle between <math>\overline{z}</math>-axis and <math>\overline{x}</math>-axis
* <math>\beta:</math> angle between <math>\overline{z}</math>-axis and  <math>\overline{y}</math>-axis
* <math>\gamma:</math> angle between <math>\overline{x}</math>-axis and <math>\overline{y}</math>-axis.

The '''angles''' can be chosen so that <math>0^\circ<\alpha+\beta<360^\circ\ .</math><br />
The '''foreshortenings''': <math>0<\; v_x,\; v_y,\; v_z\ .</math>

Only for suitable choices of angles and foreshortenings does one get undistorted images. The next diagram shows the images of the unit cube for various angles and foreshortenings and gives some hints for how to make these personal choices.

[[File:Axonometry-var14.svg|frame|none|Various axonometric images of a unit cube. (The image plane is  parallel to the y-z-plane.)<br />
The left and the far right images look more like prolonged cuboids instead of a cube.]]

[[File:Haus-karo-axonometrie.svg|thumb|Axonometry (cavalier perspective) of a house on checked pattern paper.]] 
In order to keep the drawing simple, one should choose simple foreshortenings, for example <math>1.0</math> or <math>0.5</math>.

If two foreshortenings are equal, the projection is called '''dimetric'''.<br />
If the three foreshortenings are equal, the projection is called '''isometric'''.<br />
If all foreshortenings are different, the projection is called '''trimetric'''.

The parameters in the diagram at right (e.g. of the house drawn on graph paper) are: <math>\alpha=135^\circ, \beta =90^\circ,\ v_y=v_z=1,\; v_x=1/\sqrt{2}\ .</math> Hence it is a ''dimetric'' axonometry. The image plane is parallel to the y-z-plane and any planar figure parallel to the y-z-plane appears in its true shape.

== Special axonometries ==

{| class="wikitable"
|+ Commonly used projections and perspectives with angles between the transformed coordinate axes (α, β, γ), angles from the horizon (α<sub>''h''</sub>, β<sub>''h''</sub>, without γ<sub>''h''</sub> = 90°) and scaling factors (''v<sub>i</sub>'')
! Name or property
! α = ∠''x̄z̄'' !! β = ∠''ȳz̄'' !! γ = ∠''x̄ȳ''  !! α<sub>''h''</sub> !! β<sub>''h''</sub>
! ''v<sub>x</sub>'' !! ''v<sub>y</sub>'' !! ''v<sub>z</sub>'' !! ''v''
|-
!title="plane, elevation, front-view, top-view, side-view, multiview"| Orthogonal, orthographic, planar
| 90° || 0° || 270° || 0° || 270° 
|colspan=2| ''v'' || 0% || {{n/a|any}}
|-
! Trimetric
|rowspan=6| 90° + α<sub>''h''</sub> ||rowspan=5| 90° + β<sub>''h''</sub> ||rowspan=5| 360° − α − β ||rowspan=4 {{n/a|any}} ||rowspan=4 {{n/a|any}}
|rowspan=2 {{n/a|any}}|| {{n/a|any}}|| {{n/a|any}}||rowspan=3 {{n/a|any}}
|-
! Dimetric
|colspan=2| ''v'' 
|-
! Isometric
|rowspan=2 colspan=3| ''v'' 
|-
! Normal
| 100%
|-
! Oblique, clinographic
| < 90° || < 90°
|rowspan=3 {{n/a|any}} ||rowspan=3 {{n/a|any}} ||rowspan=3 {{n/a|any}} || tan(α<sub>''h''</sub>)
|-
! Symmetric<!--?-->
| α || 360° − 2·α || < 90° || α<sub>''h''</sub>
|rowspan=2 {{n/a|any}}
|-
! Equiangular<!--?-->
|rowspan=3 colspan=3| 120° ||rowspan=3 colspan=2| 30°
|-
! Normal, 1:1 isometric
|rowspan=3 colspan=3| ''v'' || 100%
|-
! Standard, shortened isometric
| <math display=inline>\sqrt{\frac{2}{3}}</math> ≈ 81%
|-
! Pixel, 1:2 isometric
|colspan=2| 116.6° || 126.9° ||colspan=2 title="arctan({{sfrac|1|2}}) ≈ 26.57° ≈ 26°33′54″"| arctan(''v'') 
| 50%
|-
! Engineering
| 131.4° || 97.2° || 131.4° ||title="≈ 41.41° ≈ 41°24′35″"| arccos({{sfrac|3|4}}) ||title="≈ 7.18° ≈ 7°10′51″"| arcsin({{sfrac|1|8}})
| 50% ||rowspan=3 colspan=2| ''v'' ||rowspan=8| 100%
|-
! Cavalier
|rowspan=2| 90° + α<sub>''h''</sub> ||rowspan=7| 90° ||rowspan=2| 270° − α ||rowspan=2 {{n/a|any}} ||rowspan=7| 0°
| {{n/a|any}} 
|-
! Cabinet, dimetric cavalier
| < 100% 
|-
! Standard, isometric cavalier
|rowspan=2| 135° ||rowspan=2| 135° ||rowspan=2| 45° 
|colspan=3| ''v'' 
|-
! Standard 1:2 cabinet
|rowspan=3| 50% ||rowspan=4 colspan=2|''v''
|-
! 30° cabinet
| 116.6° || 153.4° ||title="arctan({{sfrac|1|2}}) = arccot(2) ≈ 26.565° ≈ 26°33′54″"| arctan(''v<sub>x</sub>'')
|-
! 60° cabinet
| 153.4° || 116.6° ||title="arccot({{sfrac|1|2}}) = arctan(2) ≈ 63.435° ≈ 63°26′06″"| arccot(''v<sub>x</sub>'')
|-
! 30° cavalier
| 120° || 150° || 30° 
| {{n/a|any}} 
|-
! Aerial, bird's eye view
|colspan=2 rowspan=2| 135° ||rowspan=5| 90° ||rowspan=2 colspan=2| 45°
|colspan=2| ''v'' || {{n/a|any}} ||rowspan=2| 100%
|-
! Military
|rowspan=4 colspan=3| ''v''
|-
! Planometric
|rowspan=3| 90° + α<sub>''h''</sub> ||rowspan=3| 180° − α<sub>''h''</sub> ||rowspan=3 {{n/a|any}} ||rowspan=3| 90° − α<sub>''h''</sub>
| {{n/a|any}}
|-
! Normal planometric
| 100%
|-
! Shortened planometric
| {{sfrac|2|3}} ≈ 67%
|-
|}

[[File:Axo-parameter-spec.svg|frame|none|Parameters of special axonometries.]]

=== Engineer projection ===
In this case {{sfn|Graf|1961|p=155}}{{sfn|Stärk|1978|p=168}}
* the ''foreshortenings'' are: <math>v_x=0.5, \ v_y=v_z=1\ </math> (dimetric axonometry) and
* the ''angles'' between the axes are: <math> \alpha=132^\circ,\ \beta=97^\circ \ .</math>
These angles are marked on many German [[set square]]s.

Advantages of an engineer projection: 
* simple foreshortenings, 
* a uniformly scaled orthographic projection with scaling factor 1.06, 
* the contour of a sphere is a circle (in general, an ellipse) .
For more details: see [[:de:Axonometrie]].

=== Cavalier perspective, cabinet perspective ===
* image plane parallel to y-z-plane.
In the literature the terms "cavalier perspective" and "cabinet perspective" are not uniformly defined. The above definition is the most general one. Often, further restrictions are applied.{{sfn|Graf|1961|p=95}}{{sfn|Stärk|1978|p=159}} For example: 
:cabinet perspective: additionally choose <math>\alpha=135^\circ</math> (oblique) and <math>v_x=0.5</math> (dimetric),
:cavalier perspective: additionally choose <math>\alpha=135^\circ</math> (oblique) and <math>v_x=1</math> (isometric).

=== Birds eye view, military projection ===
* image plane parallel to x-y-plane. 
:military projection: additionally choose <math>v_z=1</math> (isometric).
Such axonometries are often used for city maps, in order to keep horizontal figures undistorted.

=== Isometric axonometry ===
[[File:Axo-iso-beisp.svg|thumb|standard isometry: cube, cuboid, house and sphere]]
(Not to be confused with an [[isometry]] between metric spaces.)

For an '''isometric axonometry''' all foreshortenings are equal. The angles can be chosen arbitrarily, but a common choice is <math>\alpha=\beta=\gamma=120^\circ</math>.

For the '''standard isometry''' or just '''isometry''' one chooses:
* <math>v_x=v_y=v_z=1</math> (all axes undistorted) 
* <math>\alpha=\beta=\gamma=120^\circ\ .</math>

The advantage of a standard isometry: 
* the coordinates can be taken unchanged, 
* the image is a scaled orthographic projection with scale factor <math>\sqrt{1.5}=1.225</math>. Hence the image has a good impression and the contour of a sphere is a circle.  
* Some computer graphic systems (for example, [[xfig]]) provide a suitable raster (see diagram) as support.
In order to prevent scaling, one can choose the unhandy foreshortenings
* <math>v_x=v_y=v_z=\sqrt{2/3}</math> (instead of 1)
and the image is an (unscaled) orthographic projection.

[[File:Axo-examples.svg|frame|none|various axonometries of a tower<br />
dimetric military projection: <math>v_z=0.5, v_{x,y}=1</math>, dimetric engineering and cavalier projections: <math>v_x=0.5, v_{y,z}=1</math>, isometric axonometry: <math>v_{x,y,z}=1</math>]]

== Circles in axonometry ==
A parallel projection of a circle is in general an ellipse. An important special case occurs, if the circle's plane is parallel to the image plane&ndash;the image of the circle is then a congruent circle. In the diagram, the circle contained in the front face is undistorted. If the image of a circle is an ellipse, one can map four points on orthogonal diameters and the surrounding square of tangents and in the image parallelogram fill-in an ellipse by hand. A better, but more time consuming method consists of drawing the images of two perpendicular diameters of the circle, which are conjugate diameters of the image ellipse, determining the axes of the ellipse with [[Rytz's construction]] and [[ellipse#Drawing ellipses|drawing the ellipse]].
<gallery widths="200" heights="200">
Axonometrie-kreise-c.svg|Cavalier perspective: circles
File:Kugel-vogelp.svg|Military projection: sphere
</gallery>

== Spheres in axonometry ==
In a general axonometry of a sphere the image contour is an ellipse. 
The contour of a sphere is a circle only in an ''orthogonal'' axonometry. But, as the engineer projection and the standard isometry are scaled orthographic projections, the contour of a sphere is a circle in these cases, as well. As the diagram shows, an ellipse as the contour of a sphere might be confusing, so, if a sphere is part of an object to be mapped, one should choose an orthogonal axonometry or an engineer projection or a standard isometry.

== References ==
* {{cite book |last1=Graf |first1=Ulrich |first2=Martin |last2=Barner |title=Darstellende Geometrie |publisher=Quelle & Meyer |location=Heidelberg |date=1961 |isbn=3-494-00488-9 |ref={{harvid|Graf|1961}} }}
* {{cite book |last=Fucke |first=Kirch Nickel |title=Darstellende Geometrie |publisher=Fachbuch-Verlag |location=Leipzig |date=1998 |isbn=3-446-00778-4 }}
* {{cite book |last=Leopold |first=Cornelie |title=Geometrische Grundlagen der Architekturdarstellung |publisher=[[Kohlhammer Verlag]]|location=Stuttgart |date=2005 |isbn=3-17-018489-X }}
* {{cite book |last=Brailov |first=Aleksandr Yurievich |title=Engineering Graphics: Theoretical Foundations of Engineering Geometry for Design |publisher=Springer |date=2016 |isbn=978-3-319-29717-0 }}
* {{cite book |last=Stärk  |first=Roland |title=Darstellende Geometrie |publisher=Schöningh |date=1978 |isbn=3-506-37443-5 }}

;Notes
<references />

== External links ==
*[http://www.evlm.stuba.sk/~velichova/Geometria/PREDNASKY/aaxon.htm Orthogonal axonometry]

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[[Category:Descriptive geometry|*]]
[[Category:Graphical projections]]